LINEAR WAVES THAT EXPRESS THE SIMPLEST POSSIBLE PERIODIC STRUCTURE OF THE COMPRESSIBLE EULER EQUATIONS

doi:10.1016/S0252-9602(10)60015-X

Acta mathematica scientia,Series B ›› 2009, Vol. 29 ›› Issue (6): 1749-1766.doi: 10.1016/S0252-9602(10)60015-X

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LINEAR WAVES THAT EXPRESS THE SIMPLEST POSSIBLE PERIODIC STRUCTURE OF THE COMPRESSIBLE EULER EQUATIONS

Blake Temple, Robin Young

Department of Mathematics, |University of California, Davis, CA 95616, USA； Department of Mathematics and Statistics, |University of Massachusetts, Amherst, MA 01003, USA

Received:2009-11-06 Online:2009-11-20 Published:2009-11-20
Supported by:
Temple supported in part by NSF Applied Mathematics Grant Number DMS-040-6096. Young supported in part by NSF Applied Mathematics Grant Number DMS-010-4485.

Abstract

Abstract:

In this paper we show how the simplest wave structure that balances compression and rarefaction in the nonlinear compressible Euler
equations can be represented in a solution of the linearized compressible Euler equations. Such waves are exact solutions of the
equations obtained by linearizing the compressible Euler equations about the periodic extension of two constant states separated by
entropy jumps. Conditions on the states and the periods are derived which allow for the existence of solutions in the Fourier 1-mode.
In[3, 4, 5] it is shown that these are the simplest linearized waves such that, for almost every period, they are isolated in the kernel of the linearized operator that imposes periodicity, and such that they perturb to nearby nonlinear solutions of the compressible Euler equations that balance compression and rarefaction along characteristics in the formal sense described in[3]. Their fundamental nature thus makes them of interest in their own right.

Key words: compressible Euler, periodic solutions, conservation laws

CLC Number:

compressible Euler| periodic solutions| conservation laws

Blake Temple, Robin Young. LINEAR WAVES THAT EXPRESS THE SIMPLEST POSSIBLE PERIODIC STRUCTURE OF THE COMPRESSIBLE EULER EQUATIONS[J].Acta mathematica scientia,Series B, 2009, 29(6): 1749-1766.

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LINEAR WAVES THAT EXPRESS THE SIMPLEST POSSIBLE PERIODIC STRUCTURE OF THE COMPRESSIBLE EULER EQUATIONS

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